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FRM Part I · FRM Exam Part I

Option Sensitivity Measures: The Greeks for FRM Part 1

The Greeks measure how an option's value changes when one input moves: delta for the underlying price, gamma for delta itself, vega for volatility, theta for time and rho for interest rates. To solve questions, identify the Greek, apply its formula or sign rule, then scale by position size.

What this chapter covers

This chapter covers the option sensitivity measures, called the Greeks. Each one answers a simple question: if one input changes by a small amount and everything else stays fixed, how much does the option price change? The inputs are the underlying price, volatility, time to expiry and the risk-free rate.

You start with delta and delta hedging, because they are the base for everything else. Then you move to gamma, which measures how well a delta hedge holds up when the price moves. Vega, theta and rho follow. The chapter ends by combining the Greeks for whole portfolios and using a Taylor approximation to estimate the change in option value.

The chapter links closely to other parts of the paper. Option pricing in Valuation and Risk Models supplies the Black-Scholes-Merton inputs. Market risk measures such as VaR use delta and delta-gamma approximations for option positions. Bond duration and convexity follow the same Taylor logic, so what you learn here helps there too.

The Greeks produce numerical questions that you can get right with a clear method. They also test intuition: signs, long versus short positions, and how a Greek changes as an option moves in or out of the money. Both question types reward steady practice. With 100 questions in 4 hours, you cannot afford to rederive formulas in the exam. Know the rules cold, and these questions become fast, reliable marks.

Option Sensitivity Measures: The "Greeks": topics in the order to study them

  1. 1Option Greeks: Overview and DeltaDelta is the first and most used Greek, and it sets the vocabulary for the rest.
  2. 2Delta Hedging and Delta-Neutral PortfoliosOnce you know delta, you can apply it to build hedges, which is the most tested use.
  3. 3Gamma and Convexity of OptionsGamma explains why a delta hedge fails for larger moves and must be rebalanced.
  4. 4Vega and Volatility SensitivityVega adds the volatility dimension and builds on how gamma and option value relate.
  5. 5Theta and Time DecayTheta connects to gamma, since long gamma usually pays for time decay.
  6. 6Rho and Interest Rate SensitivityRho is the smallest Greek in most questions, so it is best learned after the main ones.
  7. 7Greeks of Portfolios and Taylor ApproximationThis topic pulls every Greek together into portfolio totals and approximate price changes.

How to prepare Option Sensitivity Measures: The "Greeks"

Learn the Greeks by building the logic first, then drill calculations until the steps are automatic.

  1. Write one line per Greek: what it measures, its sign for long calls and long puts, and where it is largest (at the money, near expiry, etc.).
  2. Learn the delta ranges: a call delta lies between 0 and 1, a put delta between -1 and 0. Practise from there to position delta by multiplying by the number of options.
  3. Work hedging problems in a fixed order: compute the position delta, then the opposite trade in the underlying, then the effect on the hedge when gamma is nonzero.
  4. Practise delta-gamma questions with the Taylor form: ΔC ≈ δ × ΔS + ½ × Γ × (ΔS)². Always apply the same price move to both terms.
  5. Add up Greeks for portfolios by summing position size times Greek for each holding. Treat short positions as negative.
  6. Do timed mixed questions on a calculator. Then review every wrong answer and note whether the error was a sign, a unit or a concept.
  7. In the last days, recite the sign and size table from memory and redo only the questions you missed.

Common mistakes in Option Sensitivity Measures: The "Greeks"

  • Forgetting the sign for short positions when summing portfolio Greeks.

    Fix: Write quantity as a signed number first (long positive, short negative), then multiply by the Greek.

  • Dropping the ½ in the gamma term of the Taylor approximation.

    Fix: Write ΔC ≈ δ × ΔS + ½ × Γ × (ΔS)² at the top of your working every time.

  • Hedging with the wrong number of shares or in the wrong direction.

    Fix: Set the hedge so that option delta plus underlying delta equals zero. Check the result by adding both parts.

  • Assuming a delta hedge stays neutral after the price moves.

    Fix: Remember that gamma measures this drift. A nonzero gamma means you must rebalance.

  • Mixing up units for vega, theta and rho.

    Fix: Read the unit in the question and scale your inputs to match before calculating.

  • Believing that more time to expiry always means a larger theta magnitude.

    Fix: Remember that decay is usually fastest for at-the-money options close to expiry, and slower for those with longer maturity.

Last-day revision: Option Sensitivity Measures: The "Greeks"

  • Delta = change in option price per unit change in the underlying price.
  • Call delta is between 0 and 1; put delta is between -1 and 0.
  • Position delta = number of options × delta per option, with a negative sign for short positions.
  • A delta-neutral position has total delta of zero, and it holds only for small price moves.
  • Gamma = change in delta per unit change in the underlying price; it is highest for at-the-money options near expiry.
  • Long options have positive gamma; short options have negative gamma.
  • Vega = change in option price per unit change in volatility; long calls and puts both have positive vega.
  • Theta = change in option value as time passes; it is usually negative for long options.
  • Rho = change in option price per change in the interest rate; it is generally positive for calls and negative for puts.
  • Delta-gamma approximation: ΔC ≈ δ × ΔS + ½ × Γ × (ΔS)².
  • Portfolio Greek = sum of (quantity × Greek) across all positions.
  • Fixing gamma or vega needs other options, because the underlying has zero gamma and zero vega.

Option Sensitivity Measures: The "Greeks" practice questions

Option Sensitivity Measures: The "Greeks" in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Option Sensitivity Measures: The "Greeks": frequently asked questions

Which Greek should I learn first for FRM Part I?

Start with delta. It is the base for hedging, and gamma is defined as the change in delta. Once delta is clear, the other Greeks are easier to place.

Do I need to memorise the Black-Scholes-Merton Greek formulas?

Focus first on what each Greek means, its sign and how it behaves. Know the formulas the curriculum presents, and check the current GARP Study Guide and Learning Objectives, since the curriculum is revised every year.

Can I use a financial calculator for Greeks questions?

Yes, for the arithmetic. Most questions need sums, products and the normal distribution values given in the question. A calculator helps you work fast on portfolio totals and Taylor approximations.

How do gamma and theta relate?

For a delta-hedged long option position, gains from gamma usually offset losses from theta when the price moves enough. Short gamma positions earn theta but lose when the price moves a lot.